Problem 24 - Algebra 2 and Trigonometry

Question

If money is invested at a rate of 5\(\%\) compounded annually, then for each dollar invested, the amount of money in an account is \(g(x),\) when \(g(x)=1.05^{x}\) after \(x\) years. a. Write the ordered pairs of the function \(\mathrm{g}\) for \(0 \leq x \leq 3\) and locate the pairs as points on a graph. The domain is the set of non- negative integers. b. Write the ordered pairs for \(\mathrm{g}^{-1}(x)\) and sketch the graph.

Step-by-Step Solution

Answer

a. Ordered pairs: (0, 1), (1, 1.05), (2, 1.1025), (3, 1.157625). b. Inverse pairs: (1, 0), (1.05, 1), (1.1025, 2), (1.157625, 3).

Step 1: Evaluate the Function for Given Years

Evaluate the function \(g(x) = 1.05^x\) for each integer \(x\) from 0 to 3. For \(x = 0\), \(g(0) = 1.05^0 = 1\). For \(x = 1\), \(g(1) = 1.05^1 = 1.05\). For \(x = 2\), \(g(2) = 1.05^2 = 1.1025\). For \(x = 3\), \(g(3) = 1.05^3 = 1.157625\). Thus, the ordered pairs are: \((0, 1)\), \((1, 1.05)\), \((2, 1.1025)\), \((3, 1.157625)\).

Step 2: Plot the Function Graph

Plot the points \((0, 1)\), \((1, 1.05)\), \((2, 1.1025)\), \((3, 1.157625)\) on a graph where the x-axis represents the years and the y-axis represents the function value, \(g(x)\). The curve will show how the investment grows over the years.

Step 3: Inverse Function Analysis

The inverse function \(g^{-1}(y)\) requires finding \(x\) for given \(y\) values. For this scenario, swap the x and y values from the original function. The new pairs will be \((1, 0)\), \((1.05, 1)\), \((1.1025, 2)\), \((1.157625, 3)\).

Step 4: Plot the Inverse Function Graph

Plot the points \((1, 0)\), \((1.05, 1)\), \((1.1025, 2)\), \((1.157625, 3)\) on a graph, reflecting the inverse function. The x-axis now represents \(g(x)\) values and the y-axis represents the years. This graph is the reflection of the original function across the line \(y = x\).

Key Concepts

Compound Interest

Compound interest is a powerful financial concept and it plays a crucial role in how investments grow over time. When money is invested at a compound interest rate, it earns interest not only on the initial principal amount but also on the accumulated interest from previous periods. This means your money can grow at an accelerating rate.

• Formula: The general formula for compound interest is given by \( A = P(1 + r)^n \), where \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount (the initial amount of money), \( r \) is the annual interest rate (in decimal), and \( n \) is the number of years the money is invested or borrowed.
• Example: In the exercise provided, if the interest rate is 5% per annum compounded annually, and we start with \( P = 1 \), then the function modeling the growth of this investment over time is \( g(x) = 1.05^x \). Here, 1.05 represents the factor by which the investment grows every year.

Compound interest benefits those who begin investing early, as they allow their investments to compound over a longer period, thereby maximizing returns.

Inverse Functions

An inverse function essentially reverses the operation of a given function. If a function takes an input value and produces an output, the inverse function takes the output and returns to the original input.

• Importance: Inverse functions are useful for solving problems where you need to find the original input from a given output. For example, if you know the ending amount of an investment but want to know how many years the money was invested, you can use the inverse of the growth function.
• How to find: To find the inverse of a function like \( g(x) = 1.05^x \), you would switch the x and y in the function and solve for the new y. Here, it translates into finding \( x \) for a given \( y \), thus letting us have the inverse pairs like (1,0), (1.05,1), and so on.
• Graph Reflections: In the context of graphs, the inverse function graph is a reflection over the line \( y = x \). This graphical reflection can visually verify the relationship between functions and their inverses clearly.

Function Graphing

Function graphing is a method of visualizing how a function behaves. It can help you understand relationships in data and interpret the growth and changes of functions like exponential or linear ones.

• Plotting Points: To graph a function, you start by plotting points derived from the function. In our exercise, these points are the ordered pairs calculated from the investment function \( g(x) = 1.05^x \). Points like (0, 1), (1, 1.05), provide insight into how money grows over time.
• Shape of Graphs: Exponential functions, such as compound interest in this case, show a curve that increases more steeply over time. At first, the graph might seem flat, but it becomes steeper as the values on the x-axis increase, showing accelerated growth.
• Interpreting Graphs: By analyzing the function graph, one can interpret significant patterns, such as steady growth in compounded investments. It also provides a visual comparison with the inverse function graphing, showcasing the symmetry.

Whether you're plotting real-world scenarios or solving mathematical problems, function graphing serves as a fundamental tool in visualizing and analyzing numeric relationships.